Use the unique factorization for integers theorem and the definition of logarithm to prove that log3

[winter haven]

I found this answer

but I don't see the contradiction

[modern bridge]

a and b are integers, right?

[winter haven]

yes

[modern bridge]

So we have 7 to an integer power equals 3 to an integer power. See the problem yet?

[winter haven]

not really

[modern bridge]

...okay. How many factors of 3 does 7^b have?

[winter haven]

I think the answer should be none

but how do we know that

[modern bridge]

...because 7 has no factors of 3.

a | b^c => a | b```

(if a, b, c are integers)

(Actually, if a, b are integers and c is a positive integer.)